Function Operations: How do I divide f(x) = 2x^2 + x − 3 by g(x) = x − 1?

Question

Let f(x) = 2x² + x − 3 and g(x) = x − 1.

Find  and state its domain.

The answers are:

; domain is the set of all real numbers except 1

x − 1; domain is the set of all real numbers

; domain is the set of all real numbers except 

2x + 3; domain is the set of all real numbers except 1

How does one divide a function like this by another? Why does the -3 seem to have its sign flipped in two of the answers (if either of those answers are correct)?

Answer 1

One way to do it is to notice that x-1 is a factor of 2x² + x - 3.

2x² + x - 3 = (2x + 3)(x - 1). So, (2x² + x - 3) / (x - 1) = 2x + 3 Domain = all reals except 1

Another approach is to use synthetic division:

1⌋ 2 1 -3

_______2____3

2 3 0

Since the last entry in the bottom row is zero, x - 1 is a factor of 2x² + x - 3, and the entries 2 and 3 in the bottom row are the coefficients of the other factor (i.e., the other factor is 2x + 3)